# Properties

 Label 5070.2.a.bo Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + ( 3 + \beta_{1} ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + ( 3 + \beta_{1} ) q^{7} - q^{8} + q^{9} + q^{10} + ( 2 + \beta_{1} - \beta_{2} ) q^{11} + q^{12} + ( -3 - \beta_{1} ) q^{14} - q^{15} + q^{16} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{17} - q^{18} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{19} - q^{20} + ( 3 + \beta_{1} ) q^{21} + ( -2 - \beta_{1} + \beta_{2} ) q^{22} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{23} - q^{24} + q^{25} + q^{27} + ( 3 + \beta_{1} ) q^{28} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{29} + q^{30} + ( 6 - \beta_{1} + \beta_{2} ) q^{31} - q^{32} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( 4 - 4 \beta_{1} + 5 \beta_{2} ) q^{34} + ( -3 - \beta_{1} ) q^{35} + q^{36} + ( 1 + 5 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{38} + q^{40} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -3 - \beta_{1} ) q^{42} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 2 + \beta_{1} - \beta_{2} ) q^{44} - q^{45} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{46} + ( -3 + 8 \beta_{1} - 4 \beta_{2} ) q^{47} + q^{48} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{49} - q^{50} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{51} + ( -4 - 3 \beta_{1} + 2 \beta_{2} ) q^{53} - q^{54} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} + ( -3 - \beta_{1} ) q^{56} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{57} + ( -3 + 3 \beta_{1} - 6 \beta_{2} ) q^{58} + ( 9 - 7 \beta_{1} + 9 \beta_{2} ) q^{59} - q^{60} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( -6 + \beta_{1} - \beta_{2} ) q^{62} + ( 3 + \beta_{1} ) q^{63} + q^{64} + ( -2 - \beta_{1} + \beta_{2} ) q^{66} + ( 8 - 6 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{68} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{69} + ( 3 + \beta_{1} ) q^{70} + ( 4 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} - q^{72} + ( 4 - 2 \beta_{1} + 11 \beta_{2} ) q^{73} + ( -1 - 5 \beta_{1} + 2 \beta_{2} ) q^{74} + q^{75} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{76} + ( 7 + 5 \beta_{1} - 3 \beta_{2} ) q^{77} -3 \beta_{1} q^{79} - q^{80} + q^{81} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 6 - 3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 3 + \beta_{1} ) q^{84} + ( 4 - 4 \beta_{1} + 5 \beta_{2} ) q^{85} + ( 1 + 3 \beta_{1} + 3 \beta_{2} ) q^{86} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -2 - \beta_{1} + \beta_{2} ) q^{88} + ( -2 - 2 \beta_{1} - 9 \beta_{2} ) q^{89} + q^{90} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{92} + ( 6 - \beta_{1} + \beta_{2} ) q^{93} + ( 3 - 8 \beta_{1} + 4 \beta_{2} ) q^{94} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{95} - q^{96} + ( -1 + 5 \beta_{1} + 8 \beta_{2} ) q^{97} + ( -4 - 6 \beta_{1} - \beta_{2} ) q^{98} + ( 2 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 3q^{12} - 10q^{14} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} + 11q^{19} - 3q^{20} + 10q^{21} - 8q^{22} - 9q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 10q^{28} + 3q^{30} + 16q^{31} - 3q^{32} + 8q^{33} + 3q^{34} - 10q^{35} + 3q^{36} + 10q^{37} - 11q^{38} + 3q^{40} + 5q^{41} - 10q^{42} - 3q^{43} + 8q^{44} - 3q^{45} + 9q^{46} + 3q^{47} + 3q^{48} + 17q^{49} - 3q^{50} - 3q^{51} - 17q^{53} - 3q^{54} - 8q^{55} - 10q^{56} + 11q^{57} + 11q^{59} - 3q^{60} + 11q^{61} - 16q^{62} + 10q^{63} + 3q^{64} - 8q^{66} + 15q^{67} - 3q^{68} - 9q^{69} + 10q^{70} + 13q^{71} - 3q^{72} - q^{73} - 10q^{74} + 3q^{75} + 11q^{76} + 29q^{77} - 3q^{79} - 3q^{80} + 3q^{81} - 5q^{82} + 12q^{83} + 10q^{84} + 3q^{85} + 3q^{86} - 8q^{88} + q^{89} + 3q^{90} - 9q^{92} + 16q^{93} - 3q^{94} - 11q^{95} - 3q^{96} - 6q^{97} - 17q^{98} + 8q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.24698 0.445042 1.80194
−1.00000 1.00000 1.00000 −1.00000 −1.00000 1.75302 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 3.44504 −1.00000 1.00000 1.00000
1.3 −1.00000 1.00000 1.00000 −1.00000 −1.00000 4.80194 −1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.a.bo 3
13.b even 2 1 5070.2.a.bx yes 3
13.d odd 4 2 5070.2.b.w 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bo 3 1.a even 1 1 trivial
5070.2.a.bx yes 3 13.b even 2 1
5070.2.b.w 6 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}^{3} - 10 T_{7}^{2} + 31 T_{7} - 29$$ $$T_{11}^{3} - 8 T_{11}^{2} + 19 T_{11} - 13$$ $$T_{17}^{3} + 3 T_{17}^{2} - 46 T_{17} - 139$$ $$T_{31}^{3} - 16 T_{31}^{2} + 83 T_{31} - 139$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-29 + 31 T - 10 T^{2} + T^{3}$$
$11$ $$-13 + 19 T - 8 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-139 - 46 T + 3 T^{2} + T^{3}$$
$19$ $$211 - 4 T - 11 T^{2} + T^{3}$$
$23$ $$13 + 20 T + 9 T^{2} + T^{3}$$
$29$ $$189 - 63 T + T^{3}$$
$31$ $$-139 + 83 T - 16 T^{2} + T^{3}$$
$37$ $$223 - 11 T - 10 T^{2} + T^{3}$$
$41$ $$167 - 36 T - 5 T^{2} + T^{3}$$
$43$ $$127 - 60 T + 3 T^{2} + T^{3}$$
$47$ $$559 - 109 T - 3 T^{2} + T^{3}$$
$53$ $$71 + 80 T + 17 T^{2} + T^{3}$$
$59$ $$1079 - 116 T - 11 T^{2} + T^{3}$$
$61$ $$29 + 10 T - 11 T^{2} + T^{3}$$
$67$ $$1 + 12 T - 15 T^{2} + T^{3}$$
$71$ $$13 - 30 T - 13 T^{2} + T^{3}$$
$73$ $$377 - 240 T + T^{2} + T^{3}$$
$79$ $$-27 - 18 T + 3 T^{2} + T^{3}$$
$83$ $$27 + 27 T - 12 T^{2} + T^{3}$$
$89$ $$1009 - 240 T - T^{2} + T^{3}$$
$97$ $$-2561 - 289 T + 6 T^{2} + T^{3}$$